Sieve methods for odd perfect numbers

Fletcher, S.; Nielsen, Pace; Ochem, Pascal

doi:10.1090/S0025-5718-2011-02576-7

Sieve methods for odd perfect numbers
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by S. Adam Fletcher, Pace P. Nielsen and Pascal Ochem;

Math. Comp. 81 (2012), 1753-1776

DOI: https://doi.org/10.1090/S0025-5718-2011-02576-7

Published electronically: January 9, 2012

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Abstract:

Using a new factor chain argument, we show that $5$ does not divide an odd perfect number indivisible by a sixth power. Applying sieve techniques, we also find an upper bound on the smallest prime divisor. Putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range $10^{8}<p<10^{1000}$. These results are generalized to much broader situations.

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